How to add more polygons in edit mode
There are several approaches to 'add' more polygons

Subdividing

You can open the menu 'specials' by pressing W while beeing in edit-mode, where you can find 'Subdivide' and 'Subdivide Smooth'.

https://www.blender.org/manual/modeling/meshes/editing/subdividing/subdivide.html

Note: In the Toolbar (toggle with T) options will be displayed.

You can press Ctrl + R to cut edges. This will add more polygons.
https://www.blender.org/manual/modeling/meshes/editing/subdividing/loop_subdivide.html

Mesh adding

By pressing shift + A you can open up the mesh-menu, where you can select and insert a mesh.

Extruding

By pressing E you can extrude your selected vertices, edges or faces.

https://www.blender.org/manual/modeling/meshes/editing/duplicating/extrude.html

There are multiple extrude-modes that you can find pressing space and tiping 'extrude'. That will display all actions with 'extrude' in their name.

Insetting

By pressing I Takes the currently selected faces and creates an inset of them, with adjustable thickness and depth. If several faces are selected you can choose to inset them individually. 

https://www.blender.org/manual/modeling/meshes/editing/duplicating/inset.html?highlight=inset

Knife Tool

By pressing K you can open the 'Knife Tool' where you can cut faces, what will create more polygons.

https://www.blender.org/manual/modeling/meshes/editing/subdividing/knife_subdivide.html?highlight=knife

Joining

Although this can't be made in edit-mode. In Object mode select multiple meshes and join them pressing Ctrl + J